A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Lunar distance method of Chauvenet
From: Paul Hirose
Date: 2020 May 25, 21:59 -0700
From: Paul Hirose
Date: 2020 May 25, 21:59 -0700
In another thread I mentioned the lunar distance method of William Chauvenet. I think it first appeared in 1851, in the Astronomical Journal: http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1851AJ......2...24C&db_key=AST&page_ind=0&data_type=GIF&type=SCREEN_VIEW&classic=YES The American Ephemeris and Nautical Almanac for the Year 1855 contains a description in "navigator language," two examples, and tables: https://archive.org/details/americanephemer19offigoog/page/n504/mode/2up His manual of spherical astronomy (vol. 1) says it's "the shortest and simplest of the approximative methods when these are rendered sufficiently accurate by the introduction of all necessary corrections ... There are briefer methods to be found in every work on Navigation, which will (and should) be preferred in cases where only a rude approximation to the longitude is required.": https://archive.org/details/manualofspherica01chauiala/page/402/mode/2up The tables are in volume 2: https://archive.org/details/manualofspherica02chau/page/600/mode/2up Chauvenet gives two examples in the 1855 almanac. Since my objective was to evaluate his "approximate method" or more precisely, my computer implementation, I calculated new simulated observations with Lunar 4.4. The angles are within a few arc minutes of the original examples. I could have used those examples unchanged, but that would introduce errors (such as the lunar ephemeris) which are not the fault of the Chauvenet method itself. The first is a morning Sun lunar. (Of course in practice the correct time and longitude would not be known.) 1855-09-07 08:07 UT1 (correct time) +10 s delta T 35.5° N 30° W (correct position) zero height of eye 75 F (24 C), air pressure 29.1 inches Hg (985 mb) observations: 49°25.66' Moon lower limb refracted altitude 5°18.87' Sun lower limb refracted altitude 43°52.71' observed lunar distance, near to near Almanac data: Moon at 8 h UT1 14.78' geocentric semidiameter 54.27' horizontal parallax +25°18.01' declination Sun at 8 h UT1 15.88' semidiameter 0.15' horizontal parallax +6°15.61' declination geocentric apparent distances (center to center) 08 h UT1 45°08.52' 09 h UT1 44°41.31' 10 h UT1 44°14.12' (Back in the day, lunar distances were typically tabulated every three hours. However, Chauvenet's method doesn't require that. My implementation accepts distances at any time interval. Also, it applies a correction for second differences, i.e., a uniform lunar distance rate is not assumed. The method of second difference correction is explained in the interpolation section of his spherical astronomy book.) Chauvenet solution: 45°05.300′ geocentric distance 8h07m06s observation time correct values: 45°05.34' distance 8m07m00s time Chauvenet solution error = +6 s. The other example in the 1855 almanac is an evening Fomalhaut lunar. 1855-08-30 05:40 UT1 (correct time) +10 s delta T 55° 20' S 120° 25' W (correct position) zero height of eye 20 F (-7 C), 31 inches Hg (1050 mb) altimeter setting observations 6°15.75' Moon lower limb refracted altitude 52°23.15' Fomalhaut refracted altitude 46°29.09' Moon far limb to Fomalhaut observed distance almanac data Moon at 6 h UT1 16.40' geocentric semidiameter 1°00.19' horizontal parallax +3°54.45' declination Fomalhaut -30°23.13' declination geocentric apparent distance from Moon center: 05 h UT1 44°59.00' 06 h UT1 45°32.55' 07 h UT1 46°06.13' Chauvenet solution: 45°21.352′ geocentric distance 5h39m59s observation time 45°21.36' correct distance 5h40m00s correct time solution error = -1 s The examples in the 1855 almanac appear to have been constructed to flatter Chauvenet's lunar method vs. methods that make no correction for nonstandard atmosphere conditions. One example is at high temperature and low barometer, the other at low temperature and high barometer. Both have one body at low altitude. Finally, there's the Sun lunar in his "Manual of Spherical and Practical Astronomy." In this case my re-calculated example attempts to match Chauvenet's angles to a tenth minute. 1856-03-10 03:12:49 UT1 (correct time) +10 s delta T 34°56.6' N 149°39.8' W zero height of eye 15.6 C (60.0 F) at observer 999.0 mb (29.50″ Hg) air pressure 50.0% relative humidity observations 52°34.0' Moon lower limb altitude 8°56.4' Sun lower limb altitude 44°37.0' Moon to Sun, near to near almanac data at 0300 UT1 Moon 16.34' geocentric semidiameter 59.97' horizontal parallax +14°15.18' declination Sun 16.09' geocentric semidiameter 0.15' horizontal parallax -4°03.32' declination geocentric apparent distance, center to center 3 h UT1 45°40.88' 4 h UT1 46°14.58' 5 h UT1 46°48.24' Chauvenet solution 45°48.098′ geocentric distance 3h12m51s observation time 45°48.08′ correct geocentric distance 3h12m49s correct time +2 s time error Chauvenet's approximate method had good accuracy in all my tests. (I have never evaluated his rigorous method.) However, I believe the most direct route to maximum accuracy in a computer solution is to seek the time and place that duplicates the three observed angles. This allows all factors that affect the topocentric angles to be applied, to the extent that they can be calculated.